A Unified Model of Quarks and Leptons with a Universal Texture Zero

Abstract

A bstract
We show that a universal texture zero in the (1,1) position of all fermionic mass matrices, including heavy right-handed Majorana neutrinos driving a type-I see-saw mechanism, can lead to a viable spectrum of mass, mixing and CP violation for both quarks and leptons, including (but not limited to) three important postdictions: the Cabibbo angle, the charged lepton masses, and the leptonic ‘reactor’ angle. We model this texture zero with a non-Abelian discrete family symmetry that can easily be embedded in a grand unified framework, and discuss the details of the phenomenology after electroweak and family symmetry breaking. We provide an explicit numerical fit to the available data and obtain excellent agreement with the 18 observables in the charged fermion and neutrino sectors with just 9 free parameters. We further show that the vacua of our new scalar familon fields are readily aligned along desired directions in family space, and also demonstrate discrete gauge anomaly freedom at the relevant scale of our effective theory.

Full text

OUTP-17-13P
DESY 17-146
A Unified Model of Quarks and Leptons with a Universal Texture Zero
Ivo de Medeiros Varzielas,1, ∗Graham G. Ross,2 , †and Jim Talbert3 , ‡
1CFTP, Departamento de F´ısica, Instituto Superior T´ecnico,
Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, U.K.
3Theory Group, Deutsches Elektronen-Synchrotron (DESY), D-22607 Hamburg, Germany
We show that a universal texture zero in the (1,1) position of all fermionic mass matrices, including
heavy right-handed Majorana neutrinos driving a type-I see-saw mechanism, can lead to a viable
spectrum of mass, mixing and CP violation for both quarks and leptons, including (but not limited
to) three important postdictions: the Cabibbo angle, the charged lepton masses, and the leptonic
‘reactor’ angle. We model this texture zero with a non-Abelian discrete family symmetry that can
easily be embedded in a grand unified framework, and discuss the details of the phenomenology after
electroweak and family symmetry breaking. We provide an explicit numerical fit to the available
data and obtain excellent agreement with the 18 observables in the charged fermion and neutrino
sectors with just 9 free parameters. We further show that the vacua of our new scalar familon fields
are readily aligned along desired directions in family space, and also demonstrate discrete gauge
anomaly freedom at the relevant scale of our effective theory.
∗Electronic address: ivo.de@udo.edu
†Electronic address: g.ross1@physics.ox.ac.uk
‡Electronic address: james.talbert@desy.de
arXiv:1710.01741v1 [hep-ph] 4 Oct 2017

2
I. INTRODUCTION
A convincing theory of fermion masses has proven elusive. Indeed, there is not even consensus as to whether the
pattern of quark, charged lepton, and neutrino masses is determined by dynamics or is anarchical in nature. This
confusion is partially driven by measurements of highly incongruent masses and mixings between the quark and
lepton sectors, and hence most attempts to explain this disparate spectrum choose a family symmetry that treats
them differently. There is little evidence for such extended structures given limited available data.
In this paper we study whether, due to an underlying see-saw mechanism generating neutrino masses, a unified and
family symmetric description of flavour is possible. Importantly, an enhanced symmetry of the system can lead to a
universal texture zero (UTZ) in all fermion mass matrices (Dirac and Majorana) that gives quantitative postdictions
for masses, mixings, and CP violation in good agreement with data, a significant advantage over anarchical schemes.
The structure of the underlying family symmetry we choose to study is motivated by the observation that neutrino
mixing is quite close to tribimaximal mixing, in which limit the neutrino mass eigenstates are given by
νa= (νµ+ντ)/√2
νb= (νe+νµ−ντ)/√3
νc= (2νe−νµ+ντ)/√6
(I.1)
This pattern follows if there is a Z2×Z2discrete group closed, in the νa,b,c basis, by the generators1
S=Diag(−1,−1,1), U =Diag(1,−1,−1)
which form a subgroup of S4or, more generally, of an SU(3) family symmetry. However, it is necessary that the
symmetry be broken to obtain an acceptable value of the leptonic ‘reactor’ angle θl
13. More significantly, the symmetry
must be strongly broken in the quark and charged lepton sectors where the heaviest states are mainly aligned along
the third generation, leaving an approximate SU (2) symmetry. To obtain a universal description of fermions it is
therefore necessary that aspects of both the Z2×Z2and approximate SU(2) symmetries be present. Many attempts
at integration have been made in the literature (see e.g. [1] for a recent review), including a host of model-independent
scans of finite groups [2–9] which impose specific breaking patterns down to the desired ‘residual’ symmetries in the
quark and/or lepton sectors. Unfortunately these scans only yield partially successful results for very large groups,
and in any event do not attempt to explain the dynamics of the purported symmetry breaking.
To achieve the desired patterns of mass and mixing we instead consider the group ∆(27) [10–13], which can be
understood as the (Z3×Z3)oZ3semi-direct product symmetry (see [14] for a detailed discussion of the group
properties of ∆(27)). Family symmetry eigenstates transform as ψj→eiαjψj(j= 1,2,3) under the first Z3with
αj= 2πj/3 and as cyclic permutations under the last Z3. As we discuss in the Appendix, spontaneous symmetry
breaking of either Z3can readily occur through triplet familon fields acquiring vacuum expectation values (vevs).
For the case the vev is in the hθ3i ∝ (0,0,1) direction the symmetry is broken to the Z3phase symmetry. Coupling
of the θ3familon to fermions can lead to fermion masses for the third generation. For the case the vev is in the
hθ123i ∝ (1,1,−1) direction, the symmetry breaks to the Z3permutation symmetry (up to a rephasing, see Appendix
A) and a mass can be generated for a combination of fermion generations, such as νb, along this direction. If there are
several familons both types of vev can arise and indeed a further familon can acquire vevs along the hθ23i ∝ (0,1,1)
direction, allowing for mass generation for fermions such as the νastate.
Given this pattern of symmetry breaking (and including an additional cyclic shaping symmetry) it is possible to
have a ubiquitous structure for the Dirac matrices describing up and down quarks, charged leptons, and neutrinos.
This gives a UTZ in the (1,1) direction that results in excellent postdictions for quark and charged lepton masses.
The light neutrino mass and mixing structure is quite different because the right-handed neutrinos can also have large
Majorana masses and, for the case this is dominated by the third generation mass, sequential dominance takes place
[15–18] and the see-saw mechanism suppresses the large third generation Dirac mass matrix contribution, allowing
for the light neutrino mass eigenstates to be approximately given by eq(I.1). However, unlike previous models of this
type (see e.g. [19–21] and references therein), the (1,1) texture zero structure remains after the see-saw and leads to
a specific departure from pure tribimaximal mixing in the neutrino sector, and thus gives a non-zero θl
13.
The organisation of the paper is as follows: In Section II we discuss the Dirac matrix structure needed to get
acceptable masses and mixings for the charged fermions, concentrating on the maximally symmetric form. We show
how, with an additional shaping symmetry, the structure can be generated by coupling the fermions to the θ3, θ23
1Acting with Spermutes νa→ −νa,νb→ −νb,νc→νcand similarly for U.

3
and θ123 familons. In Section III we use the underlying symmetries to constrain both the Dirac and Majorana mass
matrices for the neutrinos, show that the (1,1) texture zero persists for the light neutrino mass matrix after a type-
I see-saw, and discuss the generic form of the resulting relations between masses and mixings. In Section IV we
explore the consistency of our model when continued to the UV by discussing the relevant discrete gauge anomalies
present, and ultimately show that our model as currently formatted is safe. Finally in Section V we show that, with
a reasonable choice of the parameters of the model, a quantitatively acceptable structure for the masses and mixing
of quarks and leptons results. The details of vacuum alignment are presented in the Appendix.
II. CHARGED FERMION MASS STRUCTURE
An important issue in the determination of fermion mass predictions is the scale at which the prediction applies. If
this is large, at the GUT or Planck scale, there will be significant radiative corrections which depend on the structure
of the theory up to this scale. In this paper we assume the scale is indeed large and further that supersymmetry2
prevents radiative corrections from driving an unacceptably large electroweak breaking scale (the hierarchy problem)
and allows for precise gauge coupling unification. In this case the radiative corrections due to gauge interactions are
well understood for the quarks. Due to the fact that the QCD coupling is much larger than the electroweak couplings,
at low scales the quark masses are enhanced by about a factor of 3 relative to the lepton masses. However this
enhancement is reduced by renormalisation group flow generated by Yukawa couplings and this introduces considerable
uncertainty due to the fact that in SUSY the top and bottom Yukawa couplings depend sensitively on tan β, the ratio
of the vevs of the two Higgs doublets in the MSSM [22–24]. In addition there may be large SUSY threshold corrections
[23].
Taking these corrections into account, quark and charged lepton masses and mixings are consistent with a symmet-
ric3mass matrix structure of the form
MD
a≈m3


0ε3
aε3
a
ε3
araε2
araε2
a
ε3
araε2
a1


, ru,d = 1/3, re=−1 (II.1)
This describes the observed masses and mixings provided the parameters a, a =u, d, e differ between the up quark
and down quark/charged lepton sectors with u≈0.05, d,e ≈0.15. This symmetric structure has a (1,1) texture
zero and, in the quark sector, implements the Gatto-Sartori-Tonin relation [25] for the Cabibbo angle given by
sin θc=rmd
ms−eiδrmu
mc
(II.2)
for some phase δ. With δ≈π/2 this is in excellent agreement with the observed masses and mixing angle. The
factors riimplement the Georgi-Jarlskog mechanism [26] giving mb=mτ, mµ= 3ms, me=1
3mdat the unification
scale, which is also in good agreement with the measured values after including radiative corrections [23].
A. Familon description
This structure can be obtained by coupling the fermions to familons θi, provided the discrete family symmetry
is supplemented by an underlying shaping symmetry. In writing the effective Lagrangian preserving the underlying
discrete ∆(27) symmetry we assume that only triplet representations are present and that the higher dimensional
operators that arise are just those consistent with the exchange of triplets, ensuring that, at the non-renormalizable
level, there are no contractions involving the non-trivial singlets of ∆(27).4The difference between the down quark
2Our flavour model does not necessarily rely on the specifics of the MSSM, and indeed the additional familons we employ are not part
of its spectrum, but of course the physical parameters we study (and ultimately fit) must be radiatively corrected to the UV in a
model-dependent way. To do this we assume the MSSM. SUSY does play a role in our vacuum alignment, cf. Appendix A.
3We are interested in the maximum symmetry consistent with all fermion masses and mixing - hence the choice of symmetric mass
matrices. An underlying SO(10) symmetry may be the origin of this structure.
4This structure is found in orbifold string compactifications [27].

4
Fields ψq,e,ν ψc
q,e,ν H5ΣS θ3θ23 θ123 θ θX
∆(27) 3 3 100 100 100 ¯
3¯
3¯
3¯
3 3
ZN0 0 0 2 -1 0 -1 2 0 x
TABLE I: Fields and their family symmetry assignments. The field θXonly plays a role in the vacuum alignment. Hence the
only requirement of its ZNcharge is that it be assigned so that the field does not contribute significantly to the fermionic mass
matrices – we have therefore left it generic.
and charged lepton matrices can be derived from an underlying GUT structure. As an example of this consider the
effective Lagrangian of the form
Leff
a,mass =ψi 1
M2
3,a
θi
3θj
3+1
M3
23,a
θi
23θj
23Σ + 1
M3
123,a
(θi
123θj
23 +θi
23θj
123)S!ψc
jH5(II.3)
where a=u, d, e and
hθ3i= v3(0,0,1),hθ23i= v23 (0,1,1)/√2,hθ123i= v123 (1,1,−1)/√3 (II.4)
The restricted form of eq(II.3) is determined by a simple ZNshaping symmetry under which the fields with non-zero
ZNare shown in Table I, along with the full symmetry assignments of our model. The field Sis ZNcharged and
indirectly affects the Majorana terms such that the UTZ is preserved (see Section III). The field Σ is associated with
the breaking of the underlying GUT with a vev ∝B−L+κT R
3. It implements the Georgi-Jarlskog relation [26] with
re/rd=−3 for κ= 0. For the case κ= 2, plus domination by the RH messengers, it gives re/rd= 3. Since the sign is
irrelevant both cases are viable. Here we concentrate on the case κ= 0 which gives rν=−1 and ru/rd= 1. Finally,
the Mi,a are the heavy mediator masses that have been integrated out when forming the effective Lagrangian. There
is a subtlety in that at least the top Yukawa coupling should not be suppressed and to do this one must take θ3/M3
large, a known issue in this type of model [28]. This is the case if θ3is the dominant contribution to the messenger
mass, and we assume here that this applies to the u, d and esectors. An alternative that solves this issue is through
the use of Higgs mediators as described in [29], although this is beyond the scope of the present paper as it requires
an entirely different set of superfields.
B. Mass matrix parameters and messenger masses
The parameters of eq(II.1) in the (2,3) block are given by
2
a=hθ23i2hΣi
M3
23,a
.M2
3,a
hθ3i2(II.5)
Referring to the ZNcharges of the fields as Q, if the Q= 0/Q =−1 mediator mass ratio M3,a
M23,a is smaller in the
up sector than in the down sector, one will have u< d. Of course equality of the down quark and charged lepton
matrix elements in the (1,2), (2,1), and (3,3) positions requires that the expansion parameters be the same in the two
sectors. This is consistent with an underlying spontaneously broken SU (2)Rsymmetry because the down quarks and
leptons are both TR,3=−1/2 states and, in SUSY, both acquire their mass from the same Higgs doublet, Hd.
Here we consider the case that the messengers carry quark and lepton quantum numbers. For the messengers
carrying left-handed quantum number, SU (2)Lrequires the up and down messenger masses should be equal. Thus
the only way the expansion parameters can be different in the up and down sectors is if the right-handed messengers
dominate. In this case, if the underlying symmetry breaking pattern is
SO(10) →SU (4) ×SU(2)L×SU (2)R→SU (3) ×SU(2)L×U(1) (II.6)
the down quarks and charged leptons will have the same expansion parameter after SU (2)Rbreaking.
Up to signs and O(1) coefficients allowed by the ZNsymmetry, the (1,j), (j,1) entries of eq(II.1) are given by
3
a=hθ23ihθ123 ihSi
M3
123,a
.M2
3,a
hθ3i2,(II.7)
to be consistent with the form of eq(II.1). Since they involve both the Q= 1 and Q=−1 mediator masses there is
sufficient freedom for this to be the case.

5
C. Higher order operators
We may also be sensitive to terms of higher mass dimension in the operator product expansion of the effective
theory. The higher order operators allowed by the symmetries in Table I at the next order relevant to contributions
in eq(II.1) are of dimension eight, and hence are suppressed by four powers of the relevant messenger masses:
LHO
a,mass =ψi 1
M4
23,a
(θi
23θj
3+θi
3θj
23)ΣS+1
M4
123,a
(θi
123θj
3+θi
3θj
123)S2!ψc
jH5(II.8)
However, the relative magnitude of their contributions are of different orders in the mass matrix. Assuming ap-
proximately universal messenger masses and given that the lowest order contributions involving the vev of Σ are
parametrically larger than those involving S, one finds that
hθ23ihθ23 ihΣi
M3
23 ∼ O(2),hθ23ihθ123ihSi
M3
123 ∼ O(3) =⇒hθ23ihΣi
hθ123ihSi∼ O 1
(II.9)
where is the small parameter of eq(II.1). The contributions ∝ΣSin eq(II.8) are therefore also parametrically larger
than those ∝S2:
hθ3ihθ23ihΣihSi
M4∼1
hθ3ihθ123ihSi2
M4(II.10)
Hence we neglect the contributions to the mass matrix generated by the S2terms in the numerical fits performed in
Section V.
III. NEUTRINO MASS STRUCTURE
The neutrino sector is not as well understood as the charged fermions, as only two mass-squared differences ∆m2
ij
and the three leptonic mixings angles θl
ij are constrained to a reasonable accuracy. A recent global fit to available
neutrino data from the NuFit Collaboration [30, 31] finds ∆m2
21 '7.5×10−5eV2, ∆m2
31 '2.524 ×10−3eV2(central
values, normal mass ordering), and a leptonic PMNS mixing matrix within the 3σconfidence level of
|VP M N S |3σ∈



(0.800 −0.844) (0.515 −0.581) (0.139 −0.155)
(0.229 −0.516) (0.438 −0.699) (0.614 −0.790)
(0.249 −0.528) (0.462 −0.715) (0.595 −0.776)



(III.1)
The leptonic CP violating phase is not constrained at the 3σconfidence level. Unlike quark mixing, leptonic mixing
is clearly large, non-hierarchical, and still consistent with tribimaximal mixing up to obvious corrections in the (1,3)
element. However, neutrinos’ fundamental nature as either Dirac or Majorana fermions, mass generation mechanism,
absolute mass values, and associated CP violating phase(s) are currently unknown. Furthermore, as with the charged
fermions, we must be concerned about radiative corrections to neutrino mass and mixing parameters. The case where
neutrino masses are generated with a type-I see-saw mechanism and radiatively corrected with an MSSM spectrum
is well studied [32–36]. The authors of [36] conclude that, while a degenerate (or nearly degenerate) mass spectrum,
large tan β, and/or special configurations of Dirac and Majorana CP violating phases can conspire and contribute
to substantive running for the mixing parameters, the general expectation is that ∆θν
ij ≡θν
ij (ΛGUT )−θν
ij (ΛMZ )∼
O(10−1−10−3), even for rather large values of tan β. Given that we predict a hierarchical mass spectrum with the
lightest neutrino mass many orders of magnitude smaller than the rest, we take the current 3σbounds from NuFit to
be valid in the UV as well.
Neutrino masses are more sensitive to radiative effects and can change by tens of percent over many decades of
evolution. In fact, in certain scenarios a normal spectrum in the UV can look like an inverted spectrum in the IR
[36]! Our solutions in the charged fermion sector tend to favor larger values of tan β, and in this scenario the heaviest
mass eigenstate will split from the lighter ones during its RGE. This means that our principal mass prediction, the
ratio of the solar and atmospheric mass splitting, will diminish in the UV. Using the most recent values from NuFit
one finds (in the IR) that
∆m2
sol
∆m2
atm ∈ {.0266, .0336}(III.2)
although we estimate that ∆m2
sol
∆m2
atm &.021 at the GUT scale, given the above discussion.

6
A. Familon description
We again find that this generic structure can be understood by coupling neutrino family triplets to familons θi
although, due to the see-saw mechanism, the neutrino mass matrix will obviously have a different structure than
the charged fermions. In the context of an underlying SO(10) the neutrinos must have the same form of the Dirac
Lagrangian, eq(II.1). Taking the case κ= 0 gives rν=−1.
On the other hand, the Majorana mass matrix requires lepton number violation. In the context of the familon
structure introduced above it is an obvious choice to assume that the lepton number violation occurs through the
vev of a further familon triplet field θcarrying lepton number −1. Then the Lagrangian terms responsible for the
Majorana mass, consistent with the underlying ∆(27) symmetry, are given by
Lν
Maj orana mass =ψc
i1
Mθiθj+1
M4[c1θi
23θj
23(θaθaθa
123) + c2(θi
23θj
123 +θi
123θj
23)(θaθaθa
23)]ψc
j(III.3)
Due to the different mediators (and couplings) we have allowed for different coefficients c1,c2of the two components
of the second term. In this form, we note the absence of terms with two θ123 familons, which would destroy the UTZ
(the field Swhich appears in the Dirac terms only is indirectly responsible for this).
B. Qualitative analysis of neutrino masses and mixing
The high inverse power of the mediator mass associated with the second term of eq(III.3) allows the hierarchical
structure in the Majorana mass matrix to readily be much greater than that in the Dirac matrix. In this case the
contribution to the LH neutrino masses via the see-saw with νc
3exchange is negligible and thus the mass matrix
structure giving mass via the see-saw to the 2 heaviest neutrinos is effectively two dimensional. The Majorana mass
matrix is defined in the (ν1, ν2) basis and the Dirac mass matrix is in the (νb, νa)(ν1, ν2) basis where νa,b are given
in eq(I.1). In this basis (and taking κ= 0) the application of the type-I see-saw generates a simple matrix of two
complex parameters:
MMaj orana ∝
0c2
c2c1+ 2 c2
, MDirac ∝
0p3/2
1 1 + s
=⇒
|{z}
see-saw
Mν∝
0−p3/2c2
−p3/2c2c0
1
(III.4)
where c0
1≡c1−2c2swith c1c2and s∝ hΣihθ23i/(hSihθ123 i). From this one easily finds that the ratio of neutrino
masses is given by
m2
m1≈3
2
c2
2
c0,2
1
,c2
c0
1≡ |c2
c0
1|eiδ,(III.5)
defining the phase δ, and that the heaviest neutrino mass eigenstate is
ν1∝νa−eiδrm2
m1
νb(III.6)
Thus the (1,1) texture zero gives rise to the following mixing sum rules:
sin θν
13 ≈rm2
3m1
(III.7)
sin θν
23 ≈ | 1
√2−eiδ sin θν
13|(III.8)
sin θν
12 ≈1
√3(III.9)
where the νlabel indicates that only the contribution from the neutrino mixing matrix has been included. Apart from
the solar angle θν
12, it is clear that the mixing deviates from the tribimaximal form, but now with too large a value for
the reactor angle after inputing explicit experimental values for m1,2in eq(III.7). We will show in Section V that an
excellent value for θl
13 is obtained after including the contributions predicted from the charged lepton sector, which
also affect the solar and atmospheric mixing angles. While we focus on an exact numerical approach in this paper, a
detailed analytic discussion of these effects, including the relationship between δand the standard Dirac CP-violating
phase δCP , may be found in [37].

7
∆(3N2)1k,l3[k][l]
det(h2)ωk1
det(h1)ωl1
det(h0
1)ωl1
TABLE II: Determinants over the generators of ∆(3N2) where N/3∈Z, for all irreducible representations of the group. ωis
the cubic root of unity, ω3= 1, while h1,h0
1and h2simply denote the generators of the group. Finally, the k, l indices simply
indicate different irreducible representations – see [14] for a detailed discussion of the group properties of ∆(27).
IV. DISCRETE GAUGE ANOMALIES
A long-standing argument of Krauss and Wilzcek [38] holds that apparent global discrete symmetries (Abelian
Zor non-Abelian D), e.g. R-Parity in standard SUSY models or our family symmetries, must be local/gauged in
order to avoid complications with quantum gravity (wormhole) effects. Such discrete gauge symmetries should be
anomaly free and the resultant constraints for the case of Abelian discrete symmetries were determined in [39–41].
The analogous computation for non-Abelian discrete symmetries has since been formalized [14, 42, 43] with a path-
integral approach,5concluding that the only relevant anomalies in the IR assuming a fully massless spectrum are
mixed non-Abelian gauge (G) and mixed gravitational (g) anomalies:
D−G−G, D −g−g, Z −G−G, Z −g−g(IV.1)
There are no IR anomaly constraints of the form [Z]2U(1)Yand [U(1)Y]2Zbecause the corresponding discrete
charge αof any group element transformation is always defined modulo N, the order of the group element of the
transformation, and as the hypercharges of the U(1) symmetry groups can always be rescaled, one can do so such
that this modulo constraint is satisfied.
Furthermore, cubic discrete anomalies and mixed discrete anomalies of the form Z−D−Dor D−Z−Zcan be
avoided by arguing charge fractionalization in the massive particle spectrum [39–41, 43, 44].6
The authors of [14, 42, 43] conclude that the only difference between calculating the anomaly coefficient for an
Abelian ZNor non-Abelian Ddiscrete symmetry is that, in the latter case, one must calculate the Abelian coefficients
ZNifor each generator hiof D. We call the matrix representations of these elements U, and they live in some
irreducible representation of Dlabeled by d(f):
U(d) = eiα(d)=ei2π τ(d)/N (IV.2)
The condition for a discrete symmetry transformation to be anomaly-free is not uniquely determined, but is instead
only determined modulo Ni. Via a standard derivation, one can simultaneously read off the constraint on the anomaly
coefficient for Z−G−Gor D−G−G:
Z/D −G−G:X
r(f),d(f)
tr hτ(d(f))i·l(r(f))!
= 0 mod N
2(IV.3)
The notation is such that the summation is only over chiral fermions living in representations that are non-trivial
with respect to both Gand D.l(r(f)) is the Dynkin index for a fermion living in a representation r(f)of the gauge
group. It is normalized such that l(M)=1/2,1 for SU(M) and SO(M) respectively. Of course, Abelian discrete
symmetries only have singlet irreducible representations. Here it is clear that tr τ(d(f))is a charge (called δ(f)in
[43]), and from eq(IV.2) one notes that it can be written in terms of a (multi-valued) logarithm:
tr hτ(d(f))i=Nln det U(d(f))
2πi (IV.4)
For the Abelian case, tr τ(d(f))→q(f), with q(f)the standard charge of the fermion. From eq(IV.3) and eq(IV.4)
we conclude that anomalous transformations correspond to those with det U(d(f))6= 1.
5We use the notation of [43] in the equations that follow.
6Failure to satisfy the cubic constraints can give valuable information about the ultimate order required of the Zand/or Dgroups.

8
The mixed gravitational anomaly constraints are similarly straightforward and are given by:
D−g−g:X
d(f)
tr hτ(d(f))i!
= 0 mod N
2(IV.5)
Z−g−g:X
f
q(f)=X
m
q(m)·dim R(m)!
= 0 mod N
2(IV.6)
where R(m)denotes the representations of all internal symmetries and the sum is such that each representation R(m)
only appears once.
A. Anomalies in the UTZ model
Turning to our universal texture zero model, we observe from Table I that we only ever assign fields to the (anti-
)triplet or trivial singlet representations. Yet from Table II we see that determinants over these representations are
unit in ∆(27). As the summation in eq(IV.3) and eq(IV.5) is only over fields that are non-trivial with respect to both
Dand G(or just Dfor the gravitational anomalies), and since the coefficients are always ∝det(h), we can make a
strong claim: we are free of all anomalies from the triangles D−G−Gand D−g−g, regardless of the form of the
gauge group G.
This means that we only have to be concerned with Z−G−Gand Z−g−ganomalies, yet these also turn out
to be trivially met, given the effective theory we have outlined. For one, in a non-supersymmetric model, the only
contributing fermions are the triplets of quarks, charged leptons, and neutrinos. These are not charged under the ZN
shaping symmetry, and thus contribute a vanishing anomaly coefficient. For the supersymmetric case we would in
principle have to include the fermionic partners to the familons θi,S, Higgs(es) Hu,d and the additional Σ multiplet.
However, we expect these fields to be heavy at the relevant scale of our effective Lagrangians, and hence they already
‘contribute’ to the massive state contributions on the RHS of our anomaly equations.7In order to check for anomaly
cancellation above the mass scale of these supersymmetric bosons one would also have to construct the full theory
including Froggatt-Nielsen type messenger states, which is beyond the scope of our discussion.
V. QUANTITATIVE FIT TO THE DATA
We now turn to a detailed numerical analysis of the associated phenomenology. The core predictions of our model
are complex symmetric mass matrices with a universal texture zero in the (1,1) position for all fermion families. As
our model cannot determine the overall mass scale of the fermions, we work with matrices that have been rescaled by
a factor from the (3,3) position that provides the bulk of the contribution to the third (heavy) generation. For the
Dirac masses, one obtains lowest order matrices of the form
MD
i≡MD
i
c'



0a ei(α+β+γ)a ei(β+γ)
a ei(α+β+γ)(b e−iγ + 2a e−iδ)ei(2α+γ+δ)b ei(α+δ)
a ei(β+γ)b ei(α+δ)1−2a eiγ +b eiδ



(V.1)
where i∈ {u, d, e, ν}and where a0
i=v123v3hSi
√6M3
123,a
, b0
i=rav2
23hΣi
2M3
23,a , ci=v2
3
M2
3,a and ru,d,e,ν = (1,1,−3,−3)/3. The phases
α,βare the those allowed from our generic complex vacuum alignment vectors while γand δare the implicit phases
of our complex mass matrix:
a0
c=|a0
c|eiγ ≡a eiγ ,b0
c=|b0
c|eiδ ≡b eiδ (V.2)
The form of the mass matrix is the same for the heavy singlet Majorana neutrinos, but the overall mass scale is
different. We relabel the analogous free parameters as a→y,b→x,c→M(the mass scale in eq(III.3)), γ→ρ,
δ→φ, and keep the phases from vacuum alignment labeled as αand β.
7The Higgsinos are trivially charged under the family symmetries we employ and thus would not contribute regardless of the relevant
scale.

9
�μτ
[]
�τ
[]*(-)
{ -}
-0.010 -0.005 0.000 0.005 0.010
-0.10
-0.05
0.00
0.05
0.10
(a/c)e
(b/c)e
�
[]
�
[]*(-)
{- }
-0.0010 -0.0005 0.0000 0.0005 0.0010
-0.004
-0.002
0.000
0.002
0.004
(a/c)u
(b/c)u
||
[ ]
||
[ ]
||
[ ]

[>(-)]
-3-2-10123
-3
-2
-1
0
1
2
3
γd
δd
FIG. 1: Contours from our lowest order fit. TOP LEFT: Contours of the charged lepton mass fit. Black contours represent the
bounds for the ratio of mµ/mτwhereas blue contours represent those for me/mτ, both taken from [23]. The plot is at a fixed
mτ/mτ= 1. Red dashed lines represent our solution. TOP RIGHT: The same, but for up quarks. BOTTOM LEFT: The
contours of the Jarlskog Invariant over the plane of the two free phases left in this fit. The blue plane represents the minimum
JCKM allowed in [23], and it is clear that portions of the parameter space (our solutions) can fit this. BOTTOM RIGHT:
Contours of acceptable values of |Vij |CKM and the CKM Jarlskog (interior of blue circle). The red line is the Cabibbo angle,
and regions exterior to the black circle reflect acceptable values for the (1,3) element. The relative magnitudes of the (1,3) and
(3,1) elements are not successfully resolved at lowest order in our fit. Higher order corrections as discussed in the text remedy
this.
In the quark and charged lepton sectors there are (2 + 2) ×3 + 2 = 14 parameters. This reduces to 10 parameters if
we assume an underlying GUT relation in the Georgi-Jarlskog form relating the down quarks to the charged leptons.
Six of these are phases, not all of which are physical. In fact, only two phases are relevant at leading order [45], which
we take to be γdand δd, leaving only six free parameters (including two phases). Thus the 3 mixing angles and CP
violating phase in the CKM matrix as well as the four quark and two charged lepton mass ratios are determined by
just four real parameters and two phases.
The number of parameters needed in the neutrino sector is significantly reduced in the sequential limit where the νc
3
exchange contribution to the see-saw masses is negligible. There are just two parameters (including a phase) needed
in this case (cf. eq(III.4-III.6)), plus a parameter setting the scale of neutrino masses. Thus, taking into account
the contribution of the charged leptons, the leptonic mixing angles, atmospheric and solar mass differences, and
CP violating phases are determined by two real parameters and a phase. In summary, we see that both the charged
fermion and neutrino sectors are over-constrained; 18 measurable quantities are determined by nine parameters, giving
nine predictions at leading order in the operator expansion.
Having parameterized the mass matrices, one must then reliably calculate the associated mixing matrices. The
procedure we follow is enumerated below:
1. Find the matrix with columns as eigenvectors of M2≡ M · M†.

10
(1/c) ×(a, b)e(a, b)u(a, b)ν(x, y)dd
L.O. (.0042, -.0545) (-.00014, .003) (4, 11.8)×10−5(12.75, 4.055) ×10−13 N.A.
H.O. (.00416, -.0566) (-.00014, .00275) (4, 11.8)×10−5(12.75, 4.055) ×10−13 .0145
(γ, δ )e(γ, δ)u(γ, δ )ν(ρ, φ)ψd
L.O. (.13, 1.83) (0, 0) (2π/5,0) (0,−2π/5) N.A.
H.O. (0, 2) (0,0) (2π/5,0) (0,−2π/5) π
TABLE III: Free parameters used for fitting the fermionic mass and mixing spectrum. As discussed in the text, only nine
parameters are relevant to constraining the low-energy flavour phenomenology at lowest order in the operator product expansion.
The mass and phase parameters of the down quarks are implied by the corresponding values for the charged leptons. The phase
ψdis analogous to δin eq(V.2). Note that the smallness of the Majorana neutrino parameters is compensated by a parameter
determining their overall mass scale, which is not determined in our model.
2. Diagonalize Mby defining ˆ
M=U†·M·U?.
3. Define P=diag e−i arg[ˆ
M11]/2, e−i arg [ˆ
M22]/2, e−i arg [ˆ
M33]/2.Ucan now be made generic by U→U0=U·P.
4. Diagonalize the combination M2by calculating U0† · M2·U0.
5. CKM matrices are now calculated as V†
U·VD, where V=U0.
6. For the leptonic mixing the only thing that changes is that M → M · MM,−1
νR· MTbecause of the see-saw.
Then VP MN S =V†
e·Vν.
We note that this procedure is consistent with unitary rotations in the Standard Model Yukawa8sector and charged-
current terms of the form:
uI
L→VUuLeI
L→VeeL(V.3)
dI
L→VDdLνI
L→VννL(V.4)
where {u, d, e, ν}Lare all left-handed family triplets.
A. Results of Numerical Fit
We have performed a fit where all of the up-quark phases are turned off and both γdand δdare left free. This
automatically also sets the corresponding phases for the charged leptons. The values of all of the free parameters
are given in Table III, and the corresponding predictions for the mass ratios are given in Table IV where we find
excellent agreement with data (we use [23] for our comparisons and do not assume specific values for tan β, threshold
corrections, etc.). Contours of these predictions are given in Figure 1 for the charged leptons and up quarks assuming
no higher order corrections as discussed in Section II C. The down quark contour is implied by the charged leptons.
Figure 1 also includes contours for both the CKM Jarlskog and acceptable bands of CKM mixing. The specific mixing
and CP violation predicted by this fit are given, using only lowest order parameters, by:
|VCKM|LO =



.974 .226 .00420
.226 .974 .0191
.00248 .0194 .9998



,JLO
CKM = 9.898 ×10−6(V.5)
|VP M N S |LO =



.823 .547 .152
.400 .499 .769
.404 .672 .621



,JLO
P MN S =−.0304 (V.6)
8Our low-energy neutrino mass term is of the Majorana form LM
ν∼¯νLMννc
L.

11
(µ=MX)me/mτmµ/mτmu/mtmc/mtmd/mbms/mb∆m2
sol/∆m2
atm
Max .00031 .061 8.91 ×10−6.0027 .0012 .021 .0336
Min .00022 .048 4.2×10−6.0021 .00035 .008 .021
L.O. .00031 .055 7.16 ×10−6.0027 .00090 .020 .0213
H.O. .00026 .049 7.89 ×10−6.0025 .0010 .020 .0213
TABLE IV: Results of the numerical mass fits described in the text
The PMNS matrix fits perfectly to the available NuFit data, and the CKM matrix is also pretty good given calculations
of the GUT scale acceptances. The Cabibbo sector in the (1,2) block is essentially perfect, the other off-diagonal
elements are of the correct order of magnitude, and the CKM Jarlskog invariant is successfully above its minimum
value of ∼9.8×10−6. On the other hand, the (2,3) and (3,2) elements are moderately low if rounding to the third
digit (vs. a ∼.021 minimum), and the (1,3) and (3,1) elements are not of the appropriate relative magnitudes to one
another, |V31| ≈ 2|V13 |, nor to the (2,3) sector after extracting the relevant Wolfenstein parameters. These issues are
naturally resolved by the higher order corrections of the theory.
1. Higher Order Corrections
The main deviations in the fit to the CKM matrix are in the smallest (3,1) and (1,3) elements. Being small they are
particularly sensitive to higher order terms in the operator expansion, so it is of interest to determine whether such
terms can give a fully consistent CKM matrix. The leading higher dimension operators were discussed in Section II C.
Assuming universal messenger masses and thereby neglecting contributions from the operator ∝S2, the remaining
terms ∝ΣSin eq(II.8) generate an independent entry analogous to eq(V.2) labeled d0/c ≡deiψ . We find that by
only turning on the d0contribution to the down (and therefore also charged-lepton) mass matrices we can resolve the
issues encountered in the CKM matrix at lowest order, but with the same number of free phases (we need one fewer
phase from the lowest order parameter set). Choosing dd=.0145 and its corresponding phase ψd=π, we find the
following patterns of mixing and CP violation:
|VCKM|HO =



.974 .226 .00307
.226 .974 .0313
.00574 .0309 .9995



,JHO
CKM = 1.665 ×10−5(V.7)
|VP M N S |HO =



.830 .536 .153
.405 .534 .742
.384 .654 .652



,JHO
P MN S =−.0311 (V.8)
The PMNS is again perfect, and the CKM matrix is now well within the GUT scale acceptances. Note also that
the relative magnitudes of all of the CKM elements are consistent with RGE effects, using acceptable values of the
Wolfenstein parameterization in [23]. The affect on the relevant mass ratios are seen in Table IV, where we again find
excellent agreement. We thus obtain successful predictions across the spectrum of fermionic mass and mixing data.
VI. SUMMARY AND CONCLUSION
Most attempts to determine the pattern of fermion masses and mixings have assumed that there are separate
symmetries describing the quark and the lepton sector in order to explain the disparate nature of quark and lepton
mixing angles. However we have stressed that this may not be the case if the neutrino masses are generated by the
see-saw mechanism. Exploiting this possibility we have constructed a viable model based on an egalitarian discrete
symmetry model where all fermions and additional familons are triplets under the finite group, here ∆(27). As a result,
the Dirac masses of both the quarks and leptons have the same form, albeit with different expansion parameters. The
model is consistent with both an underlying stage of Grand Unification and the absence of discrete family symmetry
anomalies.

12
A feature of the model is the appearance of a texture zero in the (1,1) position not only in the Dirac masses of
all sectors, but also in the Majorana mass matrix of the light neutrinos. Combined with a symmetric mass matrix
structure this leads to the successful Gatto-Sartori-Tonin relation for the Cabibbo angle. Assuming the Georgi-
Jarlskog GUT structure for the down-quark and charged lepton mass matrices, the texture zero gives an excellent
prediction for the electron mass. Finally in the neutrino sector the texture zero requires a departure from pure
tribimaximal mixing, leading to a non-zero value for θl
13 consistent with the observed value.
By performing a detailed numerical analysis, we show that the present measurements of fermion masses and mixings,
up to the uncertainties in the radiative evolution of these parameters to the UV, can be realized. Overall, with just 9
free parameters, excellent agreement is found with the 18 observables in the charged fermion and neutrino sectors. As
such it provides some evidence in favour of a dynamical rather than anarchical origin for fermion masses and mixings.
Acknowledgements
IdMV acknowledges funding from Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) through the contract
IF/00816/2015. This work was partially supported by Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT, Portugal)
through the project CFTP-FCT Unit 777 (UID/FIS/00777/2013) which is partially funded through POCTI (FEDER),
COMPETE, QREN and EU. J.T. acknowledges research and travel support from DESY. GGR thanks CERN for vis-
iting support during which part of this work was conducted.
Appendix A: Vacuum alignment
In what follows we consider the minimum number of triplet familon fields that can lead to the desired vacuum
alignment. These are the four anti-triplet fields θ3,23,123 and θintroduced above together with a fifth triplet field θX.
Assuming the underlying theory is supersymmetric we should include in the potential only those terms consistent
with (spontaneously broken) supersymmetry (SUSY). For the case the associated familon superfields are R singlets
there are no cubic terms in the superpotential involving only familon fields and hence, in the supersymmetric limit,
no quartic terms. After supersymmetry breaking the scalar components of the superfields acquire SUSY breaking
masses, giving the potential
V1(θi) = m2
i|θi|2(A.1)
Radiative corrections can drive m2
inegative, triggering spontaneous breaking [45] of the family symmetry at a scale
close to the scale at which m2
iis zero, and this may happen for all the familon fields.
These are the dominant terms that set the scale for the familon vevs. However, being SU(3)finvariant, these
terms do not align the vevs in the manner required. To do that we need to consider terms allowed by the discrete
symmetry that are not SU (3)fsymmetric. In studying this it is necessary to determine which couplings dominate.
In the context of a supersymmetric UV completion the leading quartic couplings come from F-terms associated
with trilinear couplings to heavy mediators in the superpotential and, due to F-term decoupling, the couplings are
small, suppressed by the square of the supersymmetry breaking scale over the mediator scale (m0/M)2, and depend
sensitively on the mediator spectrum. As discussed above, we allow only triplet mediators and consider the most
general set of effective couplings that can arise from the exchange of such mediators.
Consider the case that the dominant coupling for the θ3,123 fields is the self-coupling term
V2(θi) = hi(θi)2θ†i2.(A.2)
Minimising the potential9one sees that these terms align the field vevs, the direction depending on the sign of h:
hθii=



0
0
1



vθ, hi<0,hθii=1
√3



1
1
1



vθ, hi>0
These are in the directions required for θ3and θ123!
9For clarity we assume real vevs here. The general case is presented below.

13
To complete the model it is necessary to arrange the alignment of the θ23 field vev. The field θXcan readily be
made orthogonal to θ123 if its dominant effective coupling is
V3=k1θX,iθ†i
123θ123,j θ†j
X, k1>0.(A.3)
However this term does not distinguish between (0,1,−1)/√2 and (2,−1,−1)/√6 (up to permutations of the ele-
ments). The latter vev is chosen if the dominant term sensitive to the difference is
V4=k2m0θ1
Xθ2
Xθ3
X(A.4)
Although a cubic term in the superpotential involving the θXsuperfield is forbidden by R-symmetry, it is generated
with coefficient m0after SUSY breaking. Then, in supergravity, the cubic term in the potential appears with k2=
O(m0/M) where m0is the gravitino mass. With this the final alignment of θ23 is driven by the term
V5=k3θ23,iθi
Xθ†j
23θ†j
X+k4θ23,iθ†i
3θ3,iθ†i
23,with k3>0 and k4<0 (A.5)
To summarise, the potential
V=X
i=3,123
(V1(θi) + V2(θi)) + V3+V4+V5(A.6)
aligns the fields in the directions
hθ3i=



0
0
1



v3,hθ123i=1
√3



eiβ
eiα
−1



v123,hθ23 i=1
√2



0
eiα
1



v23,Dθ†
XE=1
√6



2eiβ
−eiα
1



vX(A.7)
where we have now included the relative phases explicitly. The vevs vimay also be complex. Note that further quartic
terms allowed by the symmetries may be present but they should be subdominant to preserve this alignment. It is
straightforwrd to assign a ZNcharge to θXso that it does not contribute significantly to the fermion mass matrix.10
Finally, it is necessary to align the θfamilon that carries lepton number -1. This is readily the case through the
potential
Vθ=V1(θ) + V2(θ) + k5θ3,iθ†iθiθ†i
3, k5<0 (A.8)
10 A significant contribution of θXto fermion masses can also be avoided with an R-symmetry but, as this depends on the details of the
underlying SUSY theory, we do not discuss this here. Similarly, the cubic terms in the potential may determine some of the phases in
eq(A.7) but this too depends on the details of the symmetry properties of the underlying SUSY breaking sector.

14
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